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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font>, <font color="Maroon" title="c5">a'</font>, <font color="Maroon" title="c6">b'</font>, <font color="Maroon" title="c7">c'</font>, <font color="Maroon" title="c8">d'</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds   <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c8">d'</font>,<font color="Olive" title="b1">x</font> ) implies ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c'</font> &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d'</font> ) )</span><br/>















<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E4">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds   <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c8">d'</font>,<font color="Olive" title="b1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c'</font> &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d'</font> )</span><br/>

<a NAME="E2:23"/><b>then </b><i><font color="Green" title="E5">A2</font></i>: 
 <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c8">d'</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E3:23"/><i><font color="Green" title="E6">A3</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>;<br/>
<a NAME="E4:23"/>
 <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c8">d'</font>,1
 
<b>by </b><i><a class="txt" href="polyeq_1.html#E1:23"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>
<a NAME="E5:23"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c8">d'</font>,1
 
<b>by </b><i><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></i>;<br/>
<b>then </b><i><font color="Green" title="E7">A4</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">d</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a'</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b'</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c'</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">d'</font>
<b>by </b><i><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a'</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">b'</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">c'</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">d'</font>

;<br/>
<a NAME="E7:23"/><i><font color="Green" title="E8">A5</font></i>: 
 <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c8">d'</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="polyeq_1.html#E1:23"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>
<a NAME="E8:23"/>
3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3 = 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T2" title="POWER:th.2">POWER:2</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/>
<a NAME="E10:23"/><b>then </b><i><font color="Green" title="E9">A6</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></i>;<br/>
<a NAME="E11:23"/><i><font color="Green" title="E10">A7</font></i>: 
 <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c8">d'</font>,2
 
<b>by </b><i><a class="txt" href="polyeq_1.html#E1:23"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>
<i><font color="Green" title="E11">A8</font></i>: 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 3 = 
2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span>

;<br/>
<i><font color="Green" title="E12">A9</font></i>: 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1 = 
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> 1
<b>by </b><i><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
;<br/>
<b>thus </b><a NAME="E14:23"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c'</font> &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d'</font> )
 <b>by </b><i><a class="txt" href="polyeq_1.html#E2:23"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="polyeq_1.html#E3:23"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="polyeq_1.html#E6:23"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="polyeq_1.html#E7:23"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="polyeq_1.html#E10:23"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="polyeq_1.html#E11:23"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="polyeq_1.html#E12:23"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="polyeq_1.html#E13:23"><i><font color="Green" title="E12">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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